ATesting for the Onset of Trend, Using Wavelets

This paper develops a test for the onset of a time trend, using a wavelet‐type estimator. The series level is a nonlinear function of time, with a slope that is zero initially, but non‐negative and non‐decreasing, or non‐positive and non‐increasing, beyond the onset point, permitting divergence. The series level is otherwise unspecified, and to estimate it we regress the data series on wavelet scaling functions of time, with a coefficient restriction that makes the fitted level constant until some point in the sample. Since the true onset point is unknown, we examine several such restrictions, yielding candidate onset points at equally spaced positions in the sample. We base our test on some F‐type statistics, which compare the performance of successive fitted levels. To exploit the asymmetry of the true level under the alternative, we use a weighted sum of F stastistics, with linearly increasing weight at points further in the sample. This test statistic has a nonstandard distribution, which we tabulate. Asymptotically, we show that linear weighting gives better local power than equal weighting. A simulation study confirms the power advantage of our test.